In my poor understanding of quantum physics, quantum entanglement means that certain properties of one of two 'entangled' quantum particles can lead to change over infinitely large distances when the other particles' properties are changed.

Disregarding this already mind-boggling event taking place over say 10 meters distance; how have physicists been able to demonstrate, beyond reasonable doubt, that this can take place over infinitely large distances?

For instance: have they done some of these tests between ISS and Earth perhaps?

Just as soon as we put two detector infinitely far apart.
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dmckee♦Apr 28 '11 at 21:30

Since the ISS is only around 350 km above Earth, entanglement has been shown at a reasonable fraction of this distance. Of course, this is nowhere near infinity.
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Peter Shor Apr 28 '11 at 21:31

1

Infinitely large distances should be replaced by arbitrarily large finite distances.
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Qmechanic♦Apr 28 '11 at 21:37

1

The answers below are good, but noise in the apparatus ensures that it gets increasingly difficult to construct an experiment that verifies the violation of Bell inequalities as the distance increases. The ESA experiment is not desktop technology. Beating the effects of noise makes longer range experiments progressively more expensive and requires progressively more ingenuity. We have to introduce not only more effective shielding from noise but also more sophisticated ways to allow for the effects of the noise that can't be shielded.
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Peter MorganApr 28 '11 at 22:17

2 Answers
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First of all, it is very important to note that quantum entanglement is not a spooky action-at-a-distance as Einstein once called it! It is a strong correlation of measurements that is stronger than any classical correlation could ever be. This has been experimentally verified by the so-called violation of Bell's inequalities.

Second, quantum entanglement has been successfully demonstrated for distance of some 100 km, as the Wikipedia article on quantum teleportation states (references in Journales such as Nature are given there)

Zeilinger’s experiments on the distribution of entanglement over large distances began with both free-space and fiber-based quantum communication and teleportation between laboratories located on the different sides of the river Danube. This was then extended to larger distances across the city of Vienna and most recently over 144 km between two Canary Islands, resulting in a successful demonstration that quantum communication with satellites is feasible. His dream was to bounce entangled light off of satellites in orbit.[1] This was achieved during an experiment at the Italian Matera Laser Ranging Observatory.[4]

Admittedly, that is not a demonstration at infinite distance, but then, nothing has ever been demonstrated at that length scale and we would never be able to do and to demand such a demonstration is unreasonable. But please not the different scales involved here! Usually, we think of quantum mechanics to govern the microscopic world, involving length scales of under a micrometer, i.e. $10^{-6} m$. Achieving quantum effects on a length scale of $100 km = 10^5 m$ means you span 11 orders of magnitude. You can jokingly call this "infinite for all practical purposes".

The misunderstanding is that nothing changes in the moment of your measurement. There is a classical analogue that is wrong on the details but emphasizes the main idea: Suppose you take two playing cards, one is a king and the other a queen. You shuffle them and let your friend pick one at random while you keep the other one. Now, you have the king with probability $1/2$ and the queen also with probability 1/2, but you don't know which one you have without looking. You also don't know which one your friend has. But as soon as you look at your card, you immediately know which one your friend has: If you got the king, he must have got the queen. There is no spooky action at a distance that somehow changes what the card of your friend looks like!

In quantum mechanics, it is admittedly a bit more complex, because not only would you not know what card you have without looking, it isn't even determined until you look. It is, however, possible to go through the math and the mechanisms and come up with a satisfactory explanation that does not involve faster-than-light communication and processes.

Good answer! One additional point. As you say, you can never test at infinite distances, so the best you can do is test at very large distances. When someone says "large", you should always ask "large relative to what"? In the case of these experiments, all of the natural length scales associated with the particles (wavelengths, etc.) are microscopic, so doing an experiment over 100 km is a truly enormous distance -- probably billions of times bigger than any other relevant length scale. So while this isn't infinite distance, it's very nearly as good!
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Ted BunnApr 28 '11 at 21:17

The fundamental point is that QM is not based on probability, but on probability waves (or fields) which, differently from normal statistics, exhibit interference.
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Sklivvz♦Apr 30 '11 at 16:44

@Sklivvz There's a lecture and article by Scott Aaronson where he argues that all the axioms of QM are exactly what you expect when you define a notion of probability that uses the 2-norm instead of the 1-norm
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LagerbaerSep 20 '11 at 16:46

In September 2005, the European team
aimed ESA's one-metre telescope on the
Canary Island of Tenerife toward the
Roque de los Muchachos Observatory on
the neighbouring island of La Palma,
144 kilometres away. On La Palma, a
specially built quantum optical
terminal generated entangled photon
pairs, using the SPDC process, and
then sent one photon towards Tenerife,
whilst keeping the other for
comparison.